How to Solve Quadratic Equations: Definition, 3 Forms, 5 Methods, and Examples

A quadratic equation is an equation that can be written in standard form as:

ax² + bx + c = 0, where a ≠ 0

In this equation, a, b, and c are constants, and x is the variable. The solutions of a quadratic equation are the x-values that make the equation true. These solutions are also called roots or zeros.

Key Points Here:

• What Is a Quadratic Equation?
• Quadratic Equation Standard Form and Examples
• Forms of Quadratic Equations
• How to Solve Quadratic Equations
• Quadratic Equation Practice Problems

The most common ways to solve quadratic equations are:

• Factoring;
• Taking Square Roots;
• Completing the Square;
• Quadratic Formula;
• Graphing.

What Is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, written as:

ax² + bx + c = 0, where a ≠ 0

• a – The coefficient of x²
• b – The coefficient of x
• c- The constant term
• x – The variable
• a cannot be 0. If 𝑎 = 0, then the equation is no longer quadratic because the x² term disappears.
• The solutions of a quadratic equation are called roots, zeros, or x-intercepts. These are the x-values where the graph of the quadratic function crosses the x-axis.

When Do Students Learn Quadratic Equations?

Quadratic equations are usually introduced in Algebra 1.

• Public schools: Typically taught in high school (9th–10th grade), or as early as 8th grade in accelerated tracks.
• Private/advanced programs: Sometimes introduced as early as 7th grade.
• Quadratics return in later courses such as Geometry, Algebra II, Pre-Calculus, and also appear on standardized tests like the SAT/ACT.

Why Are Quadratic Equations Challenging?

Students often struggle because:

• Multiple solving methods can feel confusing.
• Negative signs and square roots lead to mistakes.
• Graphing links algebra with geometry
• The discriminant introduces abstract cases with complex solutions.
• Set real world model by quadratic equations

What Are the Main Forms of Quadratic Equations?

Quadratic equations can be expressed in three common forms. Each highlights different properties of the parabola.

Form	Equation	What It Shows	When It’s Useful
Standard Form	𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0	Coefficients 𝑎, 𝑏, 𝑐	Using the quadratic formula; identifying coefficients
Vertex Form	𝑎(𝑥 − ℎ)² + 𝑘 = 0	Vertex at (ℎ, 𝑘)	Graphing; finding the vertex directly
Factored (Intercept) Form	𝑎(𝑥 − 𝑝)(𝑥 − 𝑞) = 0	Roots (𝑝, 𝑞)	Quickly identifying x-intercepts

How to Graph a Quadratic Function (Parabola Explained)

The graph of a quadratic equation is a U-shaped curve called a parabola. Every parabola has key features that make it easier to analyze and graph.

Key Features of a Parabola

• Parabola: The U-shaped curve formed by graphing a quadratic function.
• Vertex: The turning point of the parabola, which is the minimum (if it opens upward) or maximum (if it opens downward).
• Axis of Symmetry: A vertical line that divides the parabola into two mirror halves. If the vertex is (ℎ, 𝑘), the axis of symmetry is 𝑥 = ℎ.
• Opening Direction:
  • If 𝑎 > 0 in the quadratic equation 𝑓(𝑥) = 𝑎𝑥² + 𝑏𝑥 + 𝑐, the parabola opens upward.
  • If 𝑎 < 0, the parabola opens downward.
• Intercepts:
  • Y-intercept: The point where the parabola crosses the y-axis (set 𝑥 = 0).
  • X-intercepts (Roots): The points where the parabola crosses the x-axis. These can be found by solving the quadratic equation.

How to Solve Quadratic Equations?

There are several ways to solve a quadratic equation, but the best method depends on how the equation is written. Before solving, look at the form of the equation first. This quick table can help students choose the most efficient method.

5 Methods	Best Used When…	Example	Why It Works
Factoring	The quadratic can be easily rewritten as two binomials	x² + 5x + 6 = 0	Turns the equation into two simpler linear equations
Taking Square Roots	The equation has an isolated square term	x² = 49 or (x – 3)² = 25	Uses square roots to undo the square
Completing the Square	Factoring is difficult, or students need vertex form	x² + 6x + 2 = 0	Rewrites the equation as a perfect square
Quadratic Formula	The quadratic does not factor easily, or students need a method that always works	2x² + 3x – 2 = 0	Works for every quadratic equation in standard form
Graphing	Students need a visual estimate of the solutions	y = x² – 4x + 3	Solutions are the x-intercepts of the parabola

💡 Quick rule:

• If the equation is easy to factor, use factoring.
• If it looks like x² = k, use square roots.
• If it does not factor easily, use the quadratic formula.
• Use graphing when you want to understand or estimate the roots visually.

In most Algebra 1 problems, students first try factoring or square roots if the equation is simple. If those methods are not convenient, the quadratic formula is the most reliable choice because it can solve any quadratic equation written as ax² + bx + c = 0.

1. How to Solve Quadratic Equations by Factoring?

Factoring works best when the quadratic can be rewritten as a product of two binomials. The key idea is the Zero Product Property: if ab = 0, then a = 0 or b = 0.

Example: How to Solve a Quadratic Equation by Factoring

Question:
Solve x² + 5x + 6 = 0.

Solution:
Find two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3, so:

x² + 5x + 6 = (x + 2)(x + 3)

Now set each factor equal to zero:

(x + 2)(x + 3) = 0 → x + 2 = 0 or x + 3 = 0

So:

x = -2 or x = -3

Answer:
The solutions are x = -2 and x = -3.

💡 Student Tip:
Factoring is usually the fastest method when the quadratic has simple integer factors.

2. How to Solve Quadratic Equations by Taking Square Roots?

Taking square roots works best when the equation can be written as x² = k or (x – h)² = k.

Example 1: x² = k

Question:
Solve x² = 49.

Solution:
Take the square root of both sides:

x = ±√49 = ±7

Answer:
The solutions are x = 7 and x = -7.

Example 2: binomial square

Question:
Solve (x – 3)² = 25.

Solution:
Take the square root of both sides:

x – 3 = ±√25 = ±5

So:

x – 3 = 5 → x = 8
x – 3 = -5 → x = -2

Answer:
The solutions are x = 8 and x = -2.

💡 Student Tip:
Do not forget the ± sign. Square root equations often have two solutions.

3. How to Solve Quadratic Equations by Completing the Square?

Completing the square rewrites a quadratic equation into a perfect square trinomial. It is useful when factoring is difficult or when students need to rewrite a quadratic in vertex form.

Example 1: How to Complete the Square Step by Step

Question:
Solve x² + 6x + 2 = 0.

Solution:
Move the constant to the other side:

x² + 6x = -2

Take half of the x coefficient and square it. Half of 6 is 3, and 3² = 9. Add 9 to both sides:

x² + 6x + 9 = -2 + 9

Rewrite the left side as a perfect square:

(x + 3)² = 7

Take the square root of both sides:

x + 3 = ±√7

So:

x = -3 ± √7

Answer:
The solutions are x = -3 + √7 and x = -3 – √7.

4. How to Solve Quadratic Equations Using the Quadratic Formula?

The quadratic formula works for any quadratic equation in standard form:

ax² + bx + c = 0, where a ≠ 0

The quadratic formula is: x = (-b ± √(b² – 4ac)) / 2a

The part under the square root, b² – 4ac, is called the discriminant.

Why does the discriminant matter?

• If b² – 4ac > 0 → two distinct real roots
• If b² – 4ac = 0 → one repeated real root
• If b² – 4ac < 0 → two complex roots

Use this method when the quadratic does not factor easily, when the answer may include radicals, or when you want a method that always works.

Example: How to Use the Quadratic Formula

Question:
Solve 2x² + 3x – 2 = 0.

Solution:
Here, a = 2, b = 3, and c = -2. Substitute these values into the quadratic formula:

x = (-b ± √(b² – 4ac)) / 2a

x = (-3 ± √(3² – 4(2)(-2))) / 2(2)
x = (-3 ± √(9 + 16)) / 4
x = (-3 ± √25) / 4
x = (-3 ± 5) / 4

Now solve both possibilities:

x = (-3 + 5) / 4 = 1/2
x = (-3 – 5) / 4 = -2

Answer:
The solutions are x = 1/2 and x = -2.

💡 Student Tip:
The most common mistakes happen when students forget the negative sign in -b or make errors inside the square root.

5. How to Solve Quadratic Equations by Graphing?

Graphing solves a quadratic equation by showing where the parabola crosses the x-axis. These x-values are the roots, zeros, or x-intercepts.

Example: How to Estimate Roots from a Parabola

Question:
Solve x² – 4x + 3 = 0 by graphing.

Solution:
The related function is: y = x² – 4x + 3

The roots are the x-values where the graph crosses the x-axis. This quadratic can also be factored:

x² – 4x + 3 = (x – 1)(x – 3)

So the x-intercepts are: x = 1 and x = 3

Answer:
The solutions are x = 1 and x = 3.

💡 Student Tip:
Graphing is helpful for visual understanding and estimating roots, but algebraic methods are usually better for exact answers.

Practice Time: Try Three Ways to Solve the Same Quadratic

Solve: x² – 2x – 3 = 0

Strengthen middle school math skills with focused practice.

Explore our all-grade worksheet hub for printable practice covering Math 6, Pre-Algebra, Algebra 1, Geometry, and Algebra 2. These resources help students review key concepts, close learning gaps, and build confidence for more advanced coursework.

7 Common Mistakes When Solving Quadratic Equations

Quadratic equations often involve multiple steps, so small errors can change the final answer. Here are the most common mistakes students should watch for.

Mistake 1: Not Setting the Equation Equal to Zero

Factoring and the quadratic formula usually require the equation to be in standard form: ax² + bx + c = 0

Incorrect: x² + 5x = -6

Trying to factor before moving all terms to one side can cause mistakes.

Correct: x² + 5x + 6 = 0

Mistake 2: Forgetting the ± Sign

When taking square roots, students often write only the positive solution.

Incorrect: x² = 25, x = 5

Correct: x² = 25, x = ±5

So: x = 5 or x = -5

Mistake 3: Misidentifying a, b, and c

In the quadratic formula, students must correctly identify a, b, and c from standard form.

For: 3x² – 7x + 2 = 0

the values are:

• a = 3
• b = -7
• c = 2

The negative sign belongs to b.

Mistake 4: Making Sign Errors in the Quadratic Formula

The quadratic formula is: x = (-b ± √(b² – 4ac)) / 2a

The formula begins with: -b

If b is negative, then -b becomes positive.

For example, if b = -7, then -b = 7

This is a common place where students lose points.

Mistake 5: Using Factoring When the Equation Does Not Factor Easily

Some quadratic equations do not factor nicely with integers.

For example: x² + 6x + 2 = 0

This does not factor easily, because 2 is only the product of 1 and 2, so completing the square or the quadratic formula is better.

Mistake 6: Ignoring Complex Solutions

If the discriminant is negative, the equation has no real solutions, but it still has complex solutions.

For example: x² + 1 = 0

Then x² = -1

There are no real number solutions, but in advanced algebra, the solutions are complex:

x = i and x = -i

Mistake 7: Not Checking the Answer

Students should check solutions by substituting them back into the original equation. This helps catch factoring mistakes, sign errors, and extra solutions.

Try It Yourself: Quadratic Equation Practice Problems

Use these problems to practice different methods.

Practice Problem 1: Factoring

Solve: x² + 7x + 12 = 0

Practice Problem 2: Factoring

Solve: x² – 9x + 20 = 0

Practice Problem 3: Taking Square Roots

Solve:

x² = 64

Practice Problem 4: Taking Square Roots

Solve: (x + 2)² = 36

Practice Problem 5: Completing the Square

Solve: x² + 8x + 5 = 0

Practice Problem 6: Quadratic Formula

Solve: x² + 3x – 10 = 0

Practice Problem 7: Quadratic Formula

Solve: 2x² – 4x – 3 = 0

Practice Problem 8: Discriminant

How many real solutions does this equation have?

x² – 6x + 9 = 0

Practice Problem 9: Graphing

A parabola crosses the x-axis at x = -1 and x = 4. What are the solutions of the quadratic equation?

Answer Key

Problem	Answer
1	x = -3, x = -4
2	x = 4, x = 5
3	x = 8, x = -8
4	x = 4, x = -8
5	x = -4 ± √11
6	x = 2, x = -5
7	x = 1 ± √10 / 2
8	One repeated real solution
9	x = -1, x = 4

Additional Math Topics – Free Worksheets

More related articles in Algebra 1

• How to Multiply Polynomials: From 1 Term to 3 Terms
• Polynomial Multiplication in Algebra: Formulas, Examples, and AMC 10 Applications
• How to Divide a Polynomial by a Monomial in 2 Simple Steps
• How to Divide Polynomials in 4 Simple Steps
• Linear inequalities
• Compound inequalities

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